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simple definition at adjoint action
I added a section about adjoint action for Hopf algebras. One could list many other flavours, like for Leibniz algebras, Lie algebroids and so on.
okay, thanks. Maybe later we can add some more general abstract stuff that makes manifest how all this is just different aspects of the same thing.
For the time being I instead just added also the adjoint action of a Lie algebra on itself, just for completeness.
we can add some more general abstract stuff that makes manifest how all this is just different aspects of the same thing
Well for adjoint action you need to have some sort of inverse. So for things like Lie groupoids, Lie algebroids etc. the thing is obvious and worthy to record and discuss, and for Hopf algebras not much more difficult.
However, for quantum groupoids/noncommutative Hopf algebroids the antipode is not really a good concept, unlike for Hopf algebroids over commutative base and all the subtleties about categorical dualities, autonomous property etc. come into play. So this may be the difficult and true research task to unify.
However, for quantum groupoids/noncommutative Hopf algebroids
I think I know how to do adjoint action of $\infty$-Lie algebroids on themselves. And I am still hoping we can understand quantum groups/quantum groupoids as equivalent re-encoding of certian Lie 2-algebras or the like. But not sure.
The jump from Hopf algebras to general Hopf algebroids over a noncommutative base is rather huge. While Drinfeld-Jimbo quantum groups are well connected to Lie theory general Hopf algebras and more general Hopf algebroids are way far. There is categorical understanding of quantum groiupoids in a famous article of Street and Day on monoidal bicategories and Hopf algebroids.
But there is an interesting remark which I do not understand in Kontsevich’s old article on Formal (non)commutative symplectic geometry, pdf. He is not yet using the lingo of “Koszul duality for operads” which exactly that paper motivated; but somewhere when he says that one can do kind of Lie calculus for various kinds of algebras, that
Another quite different aspect of duality is a kind of Lie theory. On the tensor product V ⊗ U of A-algebra V and A-dual-algebra U there is a canonical structure of Lie algebra. Category of A-dual-algebras is (more or less) equivalent to the category of functors {A-algebras} → {Lie algebras} preserving limits. At the moment we don’t understand why the homotopy theory gives the same duality as the Lie theory.
So can somebody explain to me the statement
Category of A-dual-algebras is (more or less) equivalent to the category of functors {A-algebras} → {Lie algebras} preserving limits.
So can somebody explain to me the statement
By the way, am I guessing right that “A-dual-algeba” means an algebra over the dual of the noncommutsative ring $A$?
So the claim is that every $A$-dual-algebra $U$ gives a functor
$U \otimes (-) : A-Algebras \to LieAlgebras$using for each $A$-algebra $V$ that Lie algebra structure on $U \otimes V$.
Next the claim is that this functor preserves limits (maybe finite ones?) and that every functor which preserves limits is of this form.
I do not know what do you mean about "dual of a noncommutative ring", but there is no discussion of noncommutativity particularly here, or rings in particular: the algebra is here in the sense of algebra over a dg-operad (I have put the pdf link above in 7!) so dg-algebra, dg-Lie algebra etc. are examples and the corresponding homotopy algebras are considered in order to create the calculus. The duality is probably the Koszul duality for dg-operads, but the statement is still misterious to me. The Koszul dual of a commutative algebra operad is the Lie algebra operad, cf. comparing Chevalley-Eilenberg cochain complex with the Lie algebra. Now instead of working complicated version of Lie theory for each case, the alluded fact (what does it mean "more or less true" here ?) seems to simplify and reduce other cases of Lie theory (hence maybe of integration) somehow to Lie theory in a category of functors. This is my guess, but I want that somebody tells me exactly what is going on.
“dual of a noncommutative ring”
I meant the opposite ring!
The duality is probably the Koszul duality for dg-operads
Oh, i completely missed that bit.
I’ve just written opposite ring, in case it is relevant and somebody wants to link to it. And if it’s not relevant, it’s still there!
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